Symmetries And Recursion Operators For Classical And Supersymmetric Differential Equations

Author : I.S. Krasil'shchik,P.H. Kersten
Publisher : Springer
Page : 384 pages
File Size : 49,6 Mb
Release : 2013-01-18
Category : Mathematics
ISBN : 9401731977

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Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations by I.S. Krasil'shchik,P.H. Kersten Pdf

To our wives, Masha and Marian Interest in the so-called completely integrable systems with infinite num ber of degrees of freedom was aroused immediately after publication of the famous series of papers by Gardner, Greene, Kruskal, Miura, and Zabusky [75, 77, 96, 18, 66, 19J (see also [76]) on striking properties of the Korteweg-de Vries (KdV) equation. It soon became clear that systems of such a kind possess a number of characteristic properties, such as infinite series of symmetries and/or conservation laws, inverse scattering problem formulation, L - A pair representation, existence of prolongation structures, etc. And though no satisfactory definition of complete integrability was yet invented, a need of testing a particular system for these properties appeared. Probably one of the most efficient tests of this kind was first proposed by Lenard [19]' who constructed a recursion operator for symmetries of the KdV equation. It was a strange operator, in a sense: being formally integro-differential, its action on the first classical symmetry (x-translation) was well-defined and produced the entire series of higher KdV equations; but applied to the scaling symmetry, it gave expressions containing terms of the type J u dx which had no adequate interpretation in the framework of the existing theories. It is not surprising that P. Olver wrote "The de duction of the form of the recursion operator (if it exists) requires a certain amount of inspired guesswork. . . " [80, p.

Author : Iosif Semenovich Krasilʹshchik,P.H. Kersten
Publisher : Springer Science & Business Media
Page : 404 pages
File Size : 47,6 Mb
Release : 2000-05-31
Category : Computers
ISBN : 0792363159

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Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations by Iosif Semenovich Krasilʹshchik,P.H. Kersten Pdf

This book is a detailed exposition of algebraic and geometrical aspects related to the theory of symmetries and recursion operators for nonlinear partial differential equations (PDE), both in classical and in super, or graded, versions. It contains an original theory of Frölicher-Nijenhuis brackets which is the basis for a special cohomological theory naturally related to the equation structure. This theory gives rise to infinitesimal deformations of PDE, recursion operators being a particular case of such deformations. Efficient computational formulas for constructing recursion operators are deduced and, in combination with the theory of coverings, lead to practical algorithms of computations. Using these techniques, previously unknown recursion operators (together with the corresponding infinite series of symmetries) are constructed. In particular, complete integrability of some superequations of mathematical physics (Korteweg-de Vries, nonlinear Schrödinger equations, etc.) is proved. Audience: The book will be of interest to mathematicians and physicists specializing in geometry of differential equations, integrable systems and related topics.

Author : I.S. Krasil'shchik,P.H. Kersten
Publisher : Springer Science & Business Media
Page : 396 pages
File Size : 47,5 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9789401731966

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Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations by I.S. Krasil'shchik,P.H. Kersten Pdf

To our wives, Masha and Marian Interest in the so-called completely integrable systems with infinite num ber of degrees of freedom was aroused immediately after publication of the famous series of papers by Gardner, Greene, Kruskal, Miura, and Zabusky [75, 77, 96, 18, 66, 19J (see also [76]) on striking properties of the Korteweg-de Vries (KdV) equation. It soon became clear that systems of such a kind possess a number of characteristic properties, such as infinite series of symmetries and/or conservation laws, inverse scattering problem formulation, L - A pair representation, existence of prolongation structures, etc. And though no satisfactory definition of complete integrability was yet invented, a need of testing a particular system for these properties appeared. Probably one of the most efficient tests of this kind was first proposed by Lenard [19]' who constructed a recursion operator for symmetries of the KdV equation. It was a strange operator, in a sense: being formally integro-differential, its action on the first classical symmetry (x-translation) was well-defined and produced the entire series of higher KdV equations; but applied to the scaling symmetry, it gave expressions containing terms of the type J u dx which had no adequate interpretation in the framework of the existing theories. It is not surprising that P. Olver wrote "The de duction of the form of the recursion operator (if it exists) requires a certain amount of inspired guesswork. . . " [80, p.

Author : Joseph Krasil'shchik,Alexander Verbovetsky,Raffaele Vitolo
Publisher : Springer
Page : 263 pages
File Size : 54,5 Mb
Release : 2018-04-03
Category : Mathematics
ISBN : 9783319716558

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The Symbolic Computation of Integrability Structures for Partial Differential Equations by Joseph Krasil'shchik,Alexander Verbovetsky,Raffaele Vitolo Pdf

This is the first book devoted to the task of computing integrability structures by computer. The symbolic computation of integrability operator is a computationally hard problem and the book covers a huge number of situations through tutorials. The mathematical part of the book is a new approach to integrability structures that allows to treat all of them in a unified way. The software is an official package of Reduce. Reduce is free software, so everybody can download it and make experiments using the programs available at our website.

Author : Henrik Aratyn,Alexander S. Sorin
Publisher : Springer Science & Business Media
Page : 436 pages
File Size : 53,7 Mb
Release : 2012-12-06
Category : Science
ISBN : 9789401007207

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Integrable Hierarchies and Modern Physical Theories by Henrik Aratyn,Alexander S. Sorin Pdf

Proceedings of the NATO Advanced Research Workshop, Chicago, USA, July 22-26, 2000

Author : A.M. Vinogradov
Publisher : Springer Science & Business Media
Page : 454 pages
File Size : 50,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789400919488

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Symmetries of Partial Differential Equations by A.M. Vinogradov Pdf

2 The authors of these issues involve not only mathematicians, but also speci alists in (mathematical) physics and computer sciences. So here the reader will find different points of view and approaches to the considered field. A. M. VINOGRADOV 3 Acta Applicandae Mathematicae 15: 3-21, 1989. © 1989 Kluwer Academic Publishers. Symmetries and Conservation Laws of Partial Differential Equations: Basic Notions and Results A. M. VINOORADOV Department of Mathematics, Moscow State University, 117234, Moscow, U. S. S. R. (Received: 22 August 1988) Abstract. The main notions and results which are necessary for finding higher symmetries and conservation laws for general systems of partial differential equations are given. These constitute the starting point for the subsequent papers of this volume. Some problems are also discussed. AMS subject classifications (1980). 35A30, 58005, 58035, 58H05. Key words. Higher symmetries, conservation laws, partial differential equations, infinitely prolonged equations, generating functions. o. Introduction In this paper we present the basic notions and results from the general theory of local symmetries and conservation laws of partial differential equations. More exactly, we will focus our attention on the main conceptual points as well as on the problem of how to find all higher symmetries and conservation laws for a given system of partial differential equations. Also, some general views and perspectives will be discussed.

Author : Norbert Euler,Da-jun Zhang
Publisher : CRC Press
Page : 367 pages
File Size : 50,5 Mb
Release : 2021-09-07
Category : Mathematics
ISBN : 9781000423303

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Nonlinear Systems and Their Remarkable Mathematical Structures by Norbert Euler,Da-jun Zhang Pdf

The third volume in this sequence of books consists of a collection of contributions that aims to describe the recent progress in nonlinear differential equations and nonlinear dynamical systems (both continuous and discrete). Nonlinear Systems and Their Remarkable Mathematical Structures: Volume 3, Contributions from China just like the first two volumes, consists of contributions by world-leading experts in the subject of nonlinear systems, but in this instance only featuring contributions by leading Chinese scientists who also work in China (in some cases in collaboration with western scientists). Features Clearly illustrate the mathematical theories of nonlinear systems and its progress to both the non-expert and active researchers in this area Suitable for graduate students in Mathematics, Applied Mathematics and some of the Engineering sciences Written in a careful pedagogical manner by those experts who have been involved in the research themselves, and each contribution is reasonably self-contained

Author : A. M. Vinogradov
Publisher : American Mathematical Soc.
Page : 268 pages
File Size : 53,7 Mb
Release : 2001-10-16
Category : Mathematics
ISBN : 0821897993

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Cohomological Analysis of Partial Differential Equations and Secondary Calculus by A. M. Vinogradov Pdf

This book is dedicated to fundamentals of a new theory, which is an analog of affine algebraic geometry for (nonlinear) partial differential equations. This theory grew up from the classical geometry of PDE's originated by S. Lie and his followers by incorporating some nonclassical ideas from the theory of integrable systems, the formal theory of PDE's in its modern cohomological form given by D. Spencer and H. Goldschmidt and differential calculus over commutative algebras (Primary Calculus). The main result of this synthesis is Secondary Calculus on diffieties, new geometrical objects which are analogs of algebraic varieties in the context of (nonlinear) PDE's. Secondary Calculus surprisingly reveals a deep cohomological nature of the general theory of PDE's and indicates new directions of its further progress. Recent developments in quantum field theory showed Secondary Calculus to be its natural language, promising a nonperturbative formulation of the theory. In addition to PDE's themselves, the author describes existing and potential applications of Secondary Calculus ranging from algebraic geometry to field theory, classical and quantum, including areas such as characteristic classes, differential invariants, theory of geometric structures, variational calculus, control theory, etc. This book, focused mainly on theoretical aspects, forms a natural dipole with Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Volume 182 in this same series, Translations of Mathematical Monographs, and shows the theory "in action".

Author : W.-H. Steeb
Publisher : World Scientific
Page : 380 pages
File Size : 50,6 Mb
Release : 1996
Category : Science
ISBN : 9810228910

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Continuous Symmetries, Lie Algebras, Differential Equations, and Computer Algebra by W.-H. Steeb Pdf

This book is a comprehensive introduction to the application of continuous symmetries and their Lie algebras to ordinary and partial differential equations. It is suitable for students and research workers whose main interest lies in finding solutions to differential equations. It therefore caters for readers primarily interested in applied mathematics and physics rather than pure mathematics.The book provides an application-orientated text that is reasonably self-contained. A large number of worked examples have been included to help readers working independently of a teacher. The advance of algebraic computation has made it possible to write programs for the tedious calculations in this research field, and thus the book also makes a survey of computer algebra packages.

Author : Vladimir V Sokolov
Publisher : World Scientific
Page : 346 pages
File Size : 49,8 Mb
Release : 2020-06-05
Category : Science
ISBN : 9789811219665

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Algebraic Structures In Integrability: Foreword By Victor Kac by Vladimir V Sokolov Pdf

Relationships of the theory of integrable systems with various branches of mathematics are extremely deep and diverse. On the other hand, the most fundamental exactly integrable systems often have applications in theoretical physics. Therefore, many mathematicians and physicists are interested in integrable models.The book is intelligible to graduate and PhD students and can serve as an introduction to separate sections of the theory of classical integrable systems for scientists with algebraic inclinations. For the young, the book can serve as a starting point in the study of various aspects of integrability, while professional algebraists will be able to use some examples of algebraic structures, which appear in the theory of integrable systems, for wide-ranging generalizations.The statements are formulated in the simplest possible form. However, some ways of generalization are indicated. In the proofs, only essential points are mentioned, while for technical details, references are provided. The focus is on carefully selected examples. In addition, the book proposes many unsolved problems of various levels of complexity. A deeper understanding of every chapter of the book may require the study of more rigorous and specialized literature.

Author : Valentin Vasilʹevich Lychagin
Publisher : American Mathematical Soc.
Page : 308 pages
File Size : 53,6 Mb
Release : 1995
Category : Differential equations, Nonlinear
ISBN : 0821804286

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The Interplay between Differential Geometry and Differential Equations by Valentin Vasilʹevich Lychagin Pdf

Author : George W. Bluman,Alexei F. Cheviakov,Stephen Anco
Publisher : Springer Science & Business Media
Page : 415 pages
File Size : 42,6 Mb
Release : 2009-10-30
Category : Mathematics
ISBN : 9780387680286

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Applications of Symmetry Methods to Partial Differential Equations by George W. Bluman,Alexei F. Cheviakov,Stephen Anco Pdf

This is an acessible book on the advanced symmetry methods for differential equations, including such subjects as conservation laws, Lie-Bäcklund symmetries, contact transformations, adjoint symmetries, Nöther's Theorem, mappings with some modification, potential symmetries, nonlocal symmetries, nonlocal mappings, and non-classical method. Of use to graduate students and researchers in mathematics and physics.

Author : George W. Bluman,Sukeyuki Kumei
Publisher : Springer Science & Business Media
Page : 424 pages
File Size : 51,6 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9781475743074

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Symmetries and Differential Equations by George W. Bluman,Sukeyuki Kumei Pdf

A major portion of this book discusses work which has appeared since the publication of the book Similarity Methods for Differential Equations, Springer-Verlag, 1974, by the first author and J.D. Cole. The present book also includes a thorough and comprehensive treatment of Lie groups of tranformations and their various uses for solving ordinary and partial differential equations. No knowledge of group theory is assumed. Emphasis is placed on explicit computational algorithms to discover symmetries admitted by differential equations and to construct solutions resulting from symmetries. This book should be particularly suitable for physicists, applied mathematicians, and engineers. Almost all of the examples are taken from physical and engineering problems including those concerned with heat conduction, wave propagation, and fluid flows. A preliminary version was used as lecture notes for a two-semester course taught by the first author at the University of British Columbia in 1987-88 to graduate and senior undergraduate students in applied mathematics and physics. Chapters 1 to 4 encompass basic material. More specialized topics are covered in Chapters 5 to 7.

Author : Peter A. Clarkson,Frank W. Nijhoff
Publisher : Cambridge University Press
Page : 444 pages
File Size : 44,5 Mb
Release : 1999-02-04
Category : Mathematics
ISBN : 0521596998

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Symmetries and Integrability of Difference Equations by Peter A. Clarkson,Frank W. Nijhoff Pdf

This volume comprises state-of-the-art articles in discrete integrable systems.

Author : Tohru Morimoto,Hajime Satō,Keizo Yamaguchi
Publisher : Unknown
Page : 514 pages
File Size : 54,7 Mb
Release : 2002
Category : Mathematics
ISBN : UOM:39015057574405

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Lie Groups, Geometric Structures and Differential Equations by Tohru Morimoto,Hajime Satō,Keizo Yamaguchi Pdf

The blending of algebra, geometry, and differential equations has a long and distinguished history, dating back to the work of Sophus Lie and Elie Cartan. Overviewing the depth of their influence over the past 100 years presents a formidable challenge. A conference was held on the centennial of Lie's death to reflect upon and celebrate his pursuits, later developments, and what the future may hold. This volume showcases the contents, atmosphere, and results of that conference. Ofparticular importance are two survey articles: Morimoto develops a synthetic study of Lie groups, geometric structures, and differential equations from a unified viewpoint of nilpotent geometry. Yamaguchi and Yatsui discuss the geometry of higher order differential equations of finite type. Contributedresearch articles cover a wide range of disciplines, from geometry of differential equations, CR-geometry, and differential geometry to topics in mathematical physics. This volume is intended for graduate students studying differential geometry and analyis and advanced graduate students and researchers interested in an overview of the most recent progress in these fields. Information for our distributors: Published for the Mathematical Society of Japan by Kinokuniya, Tokyo, and distributedworldwide, except in Japan, by the AMS. All commercial channel discounts apply.